N=1 Supergravity and Maxwell superalgebras

Abstract: We present the construction of the $D=4$ supergravity action from the minimal Maxwell superalgebra $s\mathcal{M}{4}$, which can be derived from the $\mathfrak{osp}\left( 4|1\right) $ superalgebra by applying the abelian semigroup expansion procedure. We show that $N=1$, $D=4$ pure supergravity can be obtained alternatively as the MacDowell-Mansouri like action built from the curvatures of the Maxwell superalgebra $s\mathcal{M}{4}$. We extend this result to all minimal Maxwell superalgebras type $s\mathcal{M}_{m+2}$. The invariance under supersymmetry transformations is also analized.